Identifier
Values
[(1,2)] => [2,1] => [2,1] => [2,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[(1,3),(2,4)] => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[(1,4),(2,3)] => [3,4,2,1] => [2,4,3,1] => [2,4,3,1] => 2
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [2,4,3,1,6,5] => [2,4,3,1,6,5] => 3
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [6,5,3,1,2,4] => [6,5,3,1,2,4] => 2
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [4,6,3,5,1,2] => [4,6,3,5,1,2] => 3
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [5,6,3,2,1,4] => [5,6,3,2,1,4] => 2
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,4,6,5,3] => [2,1,4,6,5,3] => 3
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [5,4,3,6,1,2] => [5,4,3,6,1,2] => 3
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => 3
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation.
For a signed permutation $\sigma$, this equals
$$ \left\lfloor \dfrac{fexc(\sigma)+1}{2} \right\rfloor = exc(\sigma) + \left\lfloor \dfrac{neg(\sigma)+1}{2} \right\rfloor, $$
where
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|,$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|.$$
This statistic has the same distribution as the descent statistic St001427The number of descents of a signed permutation..
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
  • the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
  • the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
  • the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
  • the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
  • the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.
Map
to signed permutation
Description
The signed permutation with all signs positive.